# 2.1 Dsp00104-sampled time series  (Page 6/10)

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## The top two sinusoids

More important, however, is to note what has happened to the top two sinusoids. Because the frequencies of the top two sinusoids are above thefolding frequency, they no longer have a minimum of two samples per cycle. Thus, the apparent frequency of these two sinusoids has folded around the foldingfrequency and appears as a lower frequency.

## Top sinusoids match bottom sinusoids

In fact, the plot of the top sinusoid now looks exactly like the plot of the bottom sinusoid at a frequency of 0.25 cycles per second. This means that theenergy in the top sinusoid at 1.75 cycles per second has folded into a new frequency of 0.25 cycles per second.

The plot of the sinusoid immediately above the center looks exactly like the plot of the sinusoid immediately below the center at a frequency of 0.50 cyclesper second. This means that the energy in that sinusoid at 1.5 cycles per second has folded into a new frequency of 0.5 cycles per second.

## Bottom three plots are correct

The plots of the center sinusoid and the two sinusoids below the center are still correct (although not very well sampled). However, the frequency information embodied in the top two sinusoids has been lost. The top twosinusoids appear to be at a different frequency than the actual frequency of the corresponding analog signals. The fact that the frequencies of these twosinusoids were originally 1.75 and 1.50 cycles per second is now lost in the sampled data.

## A different approach

Now I'm going to illustrate the same folding phenomena from a different perspective using spectral analyses. First I will show you a case having nosampling problems. Then I will introduce a sampling problem and show you the impact that the problem has on the final results.

Figure 7 shows the result of performing spectral analyses on five different sinusoids (not the same five as in the previous discussion). Each plot in Figure 7 shows the spectrum of a different sinusoid. The spectrum is computed and displayed from zero frequency on the left to the folding frequency on theright.

Figure 7. Spectral analyses of five sinusoids with no sampling problems. ## Sampling frequency was four samples per second

The sampling frequency for the data in Figure 7 was four samples per second, giving a folding frequency of two cycles per second. Thus, the horizontal scaleon each plot represents the frequencies from zero on the left to two cycles per second on the right.

## The five sinusoids

Starting at the top, each of the five plots represents the frequency spectrum of a sinusoid having the amplitude and frequency shown in the following table.

Amplitudes and frequencies of sinusoids: ```Plot Amplitude Frequency 1 60 0.25 cycles per second2 70 0.50 cycles per second 3 80 0.75 cycles per second4 90 1.50 cycles per second 5 100 1.75 cycles per second```

## The heights of the spectral peaks

The height of each spectral peak in Figure 7 is consistent with the amplitude of the corresponding sinusoid given in the table.

what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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LITNING
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Santosh
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what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
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why we need to study biomolecules, molecular biology in nanotechnology?
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Kyle
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why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
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